Bloch's theorem infinite system?

In summary, the wavefunction becomes unphysical in an infinite system when it diverges, which means that k has to be real in order for the hamiltonian to be self-adjoint and for periodic boundary conditions to be satisfied. This is discussed in further detail in the linked article.
  • #1
thegirl
41
1
Hi,

Does anyone know why k has to be real in an infinite system for bloch's theorem. I understand that the wavefunction becomes unphysical in an infinite system as it diverges. Why does that mean k has to be real?

f(x)=u(x)exp(ikx)
 
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  • #2
Infinite systems are always quite difficult to discuss. A sound way to discuss this problem is to look at the reduced hamiltonian ##H=p^2/2m +V## acting on the functions u(x) defined on the range [0, a] where a is the length of the elementary cell. It turns out that this hamiltonian is only self-adjoint for periodic boundary conditions so that ##u(a)=u(0)\exp(ika)## where k labels all possible self-adjoint extensions. This is discussed in a pedagogical way in this article:
http://scitation.aip.org/content/aapt/journal/ajp/69/3/10.1119/1.1328351
 

FAQ: Bloch's theorem infinite system?

1. What is Bloch's theorem for infinite systems?

Bloch's theorem is a fundamental concept in solid state physics that explains the behavior of electrons in a crystal lattice. It states that the wavefunction of an electron in a periodic potential can be written as a product of a plane wave and a periodic function, known as the Bloch wavefunction.

2. How does Bloch's theorem apply to infinite systems?

Bloch's theorem can be applied to infinite systems, such as crystals, where the periodic potential extends infinitely in all directions. In this case, the Bloch wavefunction becomes a continuous function in space and can be used to describe the electronic states of the system.

3. What is the significance of Bloch's theorem for infinite systems?

Bloch's theorem allows us to understand the electronic structure of infinite systems, such as crystals, which are important in many fields of science and technology. It also provides a basis for understanding the properties of materials, such as conductivity and optical properties.

4. How is Bloch's theorem related to the band structure of materials?

Bloch's theorem is crucial in understanding the band structure of materials, which describes how electrons are allowed to occupy energy levels in a solid. The Bloch wavefunctions form the basis for constructing the energy bands and determining the electronic properties of a material.

5. Are there any limitations to Bloch's theorem for infinite systems?

While Bloch's theorem is a powerful tool for understanding the electronic structure of infinite systems, it has some limitations. It assumes a perfect periodic potential, which may not always be the case in real materials. It also does not take into account the effects of electron-electron interactions, which are important in some systems.

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